Where is the singularity in the following 3-sequential Euler Angle Rotation matrix about the ZY'Z'' axis where Y' is the new Y axis after the first Z rotation and Z'' is the new Z axis after the Y' rotation.
The matrix has been generated seen below, but I'm having trouble finding the singularity:
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Where is the singularity in the following 3-sequential Euler
Angle Rotation matrix about the ZY'Z'' axis...
In the 3D Cartesian system the rotation matrix is around the z-axis is (a 2D rotation):...
In the 3D Cartesian system the rotation matrix is around the
z-axis is (a 2D rotation):
Where
is the angle to rotate. Then rotation from A to A' is can be
represented via matrix multiplications: [A'] = [R][A]
Such a rotation is useful to return a system solved in
simplified co-ordinates to it's original co-ordinate system,
returning to original meaning to the answer. A full 3D rotation is
simply a series of 2D rotations (with the appropriate matrices)
Question: If...
If U(,) refers to a rotation through an angle ß about the y-axis, show that the...
If U(,) refers to a rotation through an angle ß about the y-axis, show that the matrix elements (j, m|U(B, Ý)|j, m'), -ism, m' Sj, are polynomials of degree 2j with respect to the variables sin (6/2) and cos (B/2). Here [j, m) refers to an eigenstate of the square and z-component of the angular momentum: j2|j, m) = jlj +1) ħaj, m), İzlj, m) = mħ|j, m).
Problem: Given a rotation R of R3 about an arbitrary axis through a given angle find the matrix w...
Problem: Given a rotation R of R3 about an arbitrary axis through a given angle find the matrix which represents R with respect to standard coordinates. Here are the details: The axis of rotation is the line L, spanned and oriented by the vector v (1,一1,-1) . Now rotate R3 about L through the angle t = 4 π according to the Right 3 Hand Rule Solution strategy: If we choose a right handed ordered ONB B- (a, b,r) for...
10. The group of rotation matrices representing rotations about the z axis by an angle a: -sin α ...
10. The group of rotation matrices representing rotations about the z axis by an angle a: -sin α 0 cos α R,(a)--| sin α cos α 0 can be viewed as a coordinate curve in SO(3). Compute the tangent vector to this curve at the identity. Similarly, find tangent vectors at the identity to the curves representing rotations about the a axis and about the y axis. Is the set of these three tangent vectors a basis for the tangent...
5. (3 pts) Any operator that transfors the same way as the position operator r under rotation is ...
5. (3 pts) Any operator that transfors the same way as the position operator r under rotation is called a vector operator. By "transforming the same way" we mean that V DV where D is the same matrix as appears in Dr. In particular for a rotation about the z axis we should have cos p-sinp0 sincos 0 0 where φ is the angle of rotation. This transformation rule follows frorn the generator of rotations where n is the unit...
A general finite rotation by an angle Φ about the axis with unit normal n, (n_k n_k = 1), is desc...
A general finite rotation by an angle Φ about the axis with unit normal n, (n_k n_k = 1), is described by ¯xi = A_ij x_j where the components of the matrix A_ij are given A_ij = cos Φ δ_ij + (1 − cos Φ) n_i n_j + sin Φ ε_ijk n_k. Evaluate A_ij A_iℓ explicitly to show that A_ij A_iℓ = δ_jℓ
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the...
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the vector obtained by rotating x counter clockwise by 7/3 around 0. Without computing any matrices, what would you expect det (T) to be? (Does T make areas larger or smaller?) Now check your answer by using the fact that the matrix for counter clockwise rotation by is cos(0) - sin(0)] A A= sin(0) cos(0) (b) Same question as (a), only this time let T...
Use the following information To help you solve the following questions. Show all work for thumbs...
Use the following information
To help you solve the following questions. Show all
work for thumbs up.
3.1 Rotations and Angular-Momentum Commutation Relations 159 We are particularly interested in an infinitesimal form of Ry: (3.1.4) where terms of order & and higher are ignored. Likewise, we have R0= ° :- R(E) = 1 (3.1.5) and (3.1.5b) - E01 which may be read from (3.1.4) by cyclic permutations of x, y, zthat is, x y , y → 2,2 → x....
This is question 5.3-5 from Introduction to Operations Research (Hillier). Relevant text: Consider the following problem....
This is question 5.3-5 from Introduction to Operations Research
(Hillier). Relevant text:
Consider the following problem. Maximize Z= cixi + c2x2 + C3X3 subject to x1 + 2x2 + x3 = b 2x1 + x2 + 3x3 = 2b and x 20, X220, X2 > 0. Note that values have not been assigned to the coefficients in the objective function (C1, C2, C3). and that the only specification for the right-hand side of the functional constraints is that the second...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
In the 3D Cartesian system the rotation matrix is around the
z-axis is (a 2D rotation):
Where
is the angle to rotate. Then rotation from A to A' is can be
represented via matrix multiplications: [A'] = [R][A]
Such a rotation is useful to return a system solved in
simplified co-ordinates to it's original co-ordinate system,
returning to original meaning to the answer. A full 3D rotation is
simply a series of 2D rotations (with the appropriate matrices)
Question: If...
If U(,) refers to a rotation through an angle ß about the y-axis, show that the matrix elements (j, m|U(B, Ý)|j, m'), -ism, m' Sj, are polynomials of degree 2j with respect to the variables sin (6/2) and cos (B/2). Here [j, m) refers to an eigenstate of the square and z-component of the angular momentum: j2|j, m) = jlj +1) ħaj, m), İzlj, m) = mħ|j, m).
Problem: Given a rotation R of R3 about an arbitrary axis through a given angle find the matrix which represents R with respect to standard coordinates. Here are the details: The axis of rotation is the line L, spanned and oriented by the vector v (1,一1,-1) . Now rotate R3 about L through the angle t = 4 π according to the Right 3 Hand Rule Solution strategy: If we choose a right handed ordered ONB B- (a, b,r) for...
10. The group of rotation matrices representing rotations about the z axis by an angle a: -sin α 0 cos α R,(a)--| sin α cos α 0 can be viewed as a coordinate curve in SO(3). Compute the tangent vector to this curve at the identity. Similarly, find tangent vectors at the identity to the curves representing rotations about the a axis and about the y axis. Is the set of these three tangent vectors a basis for the tangent...
5. (3 pts) Any operator that transfors the same way as the position operator r under rotation is called a vector operator. By "transforming the same way" we mean that V DV where D is the same matrix as appears in Dr. In particular for a rotation about the z axis we should have cos p-sinp0 sincos 0 0 where φ is the angle of rotation. This transformation rule follows frorn the generator of rotations where n is the unit...
(a) Let T: R2 + R2 be counter clockwise rotation by 7/3, i.e. T(x) is the vector obtained by rotating x counter clockwise by 7/3 around 0. Without computing any matrices, what would you expect det (T) to be? (Does T make areas larger or smaller?) Now check your answer by using the fact that the matrix for counter clockwise rotation by is cos(0) - sin(0)] A A= sin(0) cos(0) (b) Same question as (a), only this time let T...
Use the following information
To help you solve the following questions. Show all
work for thumbs up.
3.1 Rotations and Angular-Momentum Commutation Relations 159 We are particularly interested in an infinitesimal form of Ry: (3.1.4) where terms of order & and higher are ignored. Likewise, we have R0= ° :- R(E) = 1 (3.1.5) and (3.1.5b) - E01 which may be read from (3.1.4) by cyclic permutations of x, y, zthat is, x y , y → 2,2 → x....
This is question 5.3-5 from Introduction to Operations Research
(Hillier). Relevant text:
Consider the following problem. Maximize Z= cixi + c2x2 + C3X3 subject to x1 + 2x2 + x3 = b 2x1 + x2 + 3x3 = 2b and x 20, X220, X2 > 0. Note that values have not been assigned to the coefficients in the objective function (C1, C2, C3). and that the only specification for the right-hand side of the functional constraints is that the second...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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